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Theorem rexlimddvcbv 42572
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42570. The equivalent of this theorem without the bound variable change is rexlimddv 3155. Usage of this theorem is discouraged because it depends on ax-13 2371, see rexlimddvcbvw 42571 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
rexlimddvcbv.1 (𝜑 → ∃𝑥𝐴 𝜃)
rexlimddvcbv.2 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
rexlimddvcbv.3 (𝑥 = 𝑦 → (𝜃𝜒))
Assertion
Ref Expression
rexlimddvcbv (𝜑𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv
StepHypRef Expression
1 rexlimddvcbv.1 . 2 (𝜑 → ∃𝑥𝐴 𝜃)
2 rexlimddvcbv.3 . . 3 (𝑥 = 𝑦 → (𝜃𝜒))
3 rexlimddvcbv.2 . . 3 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
42, 3rexlimdvaacbv 42570 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
51, 4mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071
This theorem is referenced by: (None)
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